Question: Simplify the following expression and state the condition under which the simplification is valid: $r = \dfrac{q^2 - 13q + 30}{q^2 - 10q}$
Explanation: First factor the expressions in the numerator and denominator. $ \dfrac{q^2 - 13q + 30}{q^2 - 10q} = \dfrac{(q - 3)(q - 10)}{(q)(q - 10)} $ Notice that the term $(q - 10)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(q - 10)$ gives: $r = \dfrac{q - 3}{q}$ Since we divided by $(q - 10)$, $q \neq 10$. $r = \dfrac{q - 3}{q}; \space q \neq 10$